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(x^x)/(x+1)^(x+1)
Derivada de la función/ (x+1)/ (x^x)/ (x^x)/(x+1)^(x+1)

Derivada de (x^x)/(x+1)^(x+1)

Función f() - derivada -er orden en el punto
v

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Solución

Ha introducido [src] 
      x     
     x      
------------
       x + 1
(x + 1)
xx(x+1)x+1\frac{x^{x}}{\left(x + 1\right)^{x + 1}} 
x^x/(x + 1)^(x + 1)
Solución detallada

  1. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=xxf{\left(x \right)} = x^{x} y g(x)=(x+1)x+1g{\left(x \right)} = \left(x + 1\right)^{x + 1}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

      xx(log⁡(x)+1)x^{x} \left(\log{\left(x \right)} + 1\right)

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

      (x+1)x+1(log⁡(x+1)+1)\left(x + 1\right)^{x + 1} \left(\log{\left(x + 1 \right)} + 1\right)

    Ahora aplicamos la regla de la derivada de una divesión:

    (x+1)−2x−2(xx(x+1)x+1(log⁡(x)+1)−xx(x+1)x+1(log⁡(x+1)+1))\left(x + 1\right)^{- 2 x - 2} \left(x^{x} \left(x + 1\right)^{x + 1} \left(\log{\left(x \right)} + 1\right) - x^{x} \left(x + 1\right)^{x + 1} \left(\log{\left(x + 1 \right)} + 1\right)\right)

  2. Simplificamos:

    xx(x+1)−x−1(log⁡(x)−log⁡(x+1))x^{x} \left(x + 1\right)^{- x - 1} \left(\log{\left(x \right)} - \log{\left(x + 1 \right)}\right)

Respuesta:

xx(x+1)−x−1(log⁡(x)−log⁡(x+1))x^{x} \left(x + 1\right)^{- x - 1} \left(\log{\left(x \right)} - \log{\left(x + 1 \right)}\right)

Gráfica
02468-8-6-4-2-10102.5-2.5
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
 x        -1 - x                 x        -2 - 2*x        x + 1                 
x *(x + 1)      *(1 + log(x)) - x *(x + 1)        *(x + 1)     *(1 + log(x + 1))
−xx(x+1)−2x−2(x+1)x+1(log⁡(x+1)+1)+xx(x+1)−x−1(log⁡(x)+1)- x^{x} \left(x + 1\right)^{- 2 x - 2} \left(x + 1\right)^{x + 1} \left(\log{\left(x + 1 \right)} + 1\right) + x^{x} \left(x + 1\right)^{- x - 1} \left(\log{\left(x \right)} + 1\right)
Simplificar
Segunda derivada [src] 
 x        -1 - x /1               2                   2     1                                    \
x *(1 + x)      *|- + (1 + log(x))  + (1 + log(1 + x))  - ----- - 2*(1 + log(x))*(1 + log(1 + x))|
                 \x                                       1 + x                                  /
xx(x+1)−x−1((log⁡(x)+1)2−2(log⁡(x)+1)(log⁡(x+1)+1)+(log⁡(x+1)+1)2−1x+1+1x)x^{x} \left(x + 1\right)^{- x - 1} \left(\left(\log{\left(x \right)} + 1\right)^{2} - 2 \left(\log{\left(x \right)} + 1\right) \left(\log{\left(x + 1 \right)} + 1\right) + \left(\log{\left(x + 1 \right)} + 1\right)^{2} - \frac{1}{x + 1} + \frac{1}{x}\right)
Simplificar
Tercera derivada [src] 
 x        -1 - x /   1                   3   1                    3                      /1               2\   3*(1 + log(x))   3*(1 + log(1 + x))                  /                2     1  \\
x *(1 + x)      *|-------- + (1 + log(x))  - -- - (1 + log(1 + x))  - 3*(1 + log(1 + x))*|- + (1 + log(x)) | + -------------- + ------------------ + 3*(1 + log(x))*|(1 + log(1 + x))  - -----||
                 |       2                    2                                          \x                /         x                1 + x                         \                    1 + x/|
                 \(1 + x)                    x                                                                                                                                                 /
xx(x+1)−x−1(−3((log⁡(x)+1)2+1x)(log⁡(x+1)+1)+3((log⁡(x+1)+1)2−1x+1)(log⁡(x)+1)+(log⁡(x)+1)3−(log⁡(x+1)+1)3+3(log⁡(x+1)+1)x+1+1(x+1)2+3(log⁡(x)+1)x−1x2)x^{x} \left(x + 1\right)^{- x - 1} \left(- 3 \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) \left(\log{\left(x + 1 \right)} + 1\right) + 3 \left(\left(\log{\left(x + 1 \right)} + 1\right)^{2} - \frac{1}{x + 1}\right) \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(x \right)} + 1\right)^{3} - \left(\log{\left(x + 1 \right)} + 1\right)^{3} + \frac{3 \left(\log{\left(x + 1 \right)} + 1\right)}{x + 1} + \frac{1}{\left(x + 1\right)^{2}} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right)
Simplificar
Gráfico
Derivada de (x^x)/(x+1)^(x+1)
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